Question: $P(x)$ is a polynomial. Here are a few values of $P(x)$. $P(-5)=-2$ $P(-3)=6$ $P(3)=7$ $P(5)=-1$ What is the remainder when $P(x)$ is divided by $(x+5)$ ?
Answer: We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, the remainder when $P(x)$ is divided by $(x+5)$, which can be rewritten as $(x-({-5}))$, is $P({-5})$, and we are given that $P({-5})=-2$. In a similar manner, the remainder when $P(x)$ is divided by $(x-{3})$ is $P({3})$, and we are given that $P({3})=7$. In conclusion, The remainder when $P(x)$ is divided by $(x+5)$ is $-2$. The remainder when $P(x)$ is divided by $(x-3)$ is $7$.